On the stability of inverse dynamics control of flexible-joint parallel manipulators in the presence of modeling error and disturbances

Inverse dynamics control is considered for flexible-joint parallel manipulators in order to obtain a good trajectory tracking performance in the case of modeling error and disturbances. It is known that, in the absence of modeling error and disturbance, inverse dynamics control leads to linear fourth-order error dynamics, which is asymptotically stable if the feedback gains are chosen to make the real part of the eigenvalues of the system negative. However, when there are modeling errors and disturbances, a linear time-varying error dynamics is obtained whose stability is not assured only by keeping the real parts of the frozen-time eigenvalues of the system negative. In this paper, the stability of such systems is investigated and it is proved that the linear time-varying system can be rendered stable by selecting the feedback gains such that the variation of the system becomes sufficiently slow. To illustrate the performance of the control method, deployment motion of a 3-RPR planar parallel manipulator subject to impact is simulated. For the impact model, the impulse-momentum and the coefficient of restitution equations for the system are derived.